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Hongguang Liu - One of the best experts on this subject based on the ideXlab platform.
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effective dynamics from coherent state path Integral of full loop quantum gravity
Physical Review D, 2020Co-Authors: Muxin Han, Hongguang LiuAbstract:A new routine is proposed to relate loop quantum cosmology (LQC) to loop quantum gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path Integral Formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path Integral Formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in the ${\ensuremath{\mu}}_{0}$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\overline{\ensuremath{\mu}}$-scheme of effective dynamics and the continuum limit of the path Integral Formula.
Sonia Mazzucchi - One of the best experts on this subject based on the ideXlab platform.
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A Rigorous Mathematical Construction of Feynman Path Integrals for the Schrödinger Equation with Magnetic Field
Communications in Mathematical Physics, 2020Co-Authors: Sergio Albeverio, N. Cangiotti, Sonia MazzucchiAbstract:A Feynman path Integral Formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory Integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the Integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich Integral in the path Integral Formula for both the Schrödinger and heat equation with magnetic field.
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a rigorous mathematical construction of feynman path Integrals for the schr odinger equation with magnetic field
arXiv: Mathematical Physics, 2019Co-Authors: Sergio Albeverio, N. Cangiotti, Sonia MazzucchiAbstract:A Feynman path Integral Formula for the Schrodinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory Integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the Integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich Integral in the path Integral Formula for both the Schrodinger and heat equation with magnetic field.
Muxin Han - One of the best experts on this subject based on the ideXlab platform.
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effective dynamics from coherent state path Integral of full loop quantum gravity
Physical Review D, 2020Co-Authors: Muxin Han, Hongguang LiuAbstract:A new routine is proposed to relate loop quantum cosmology (LQC) to loop quantum gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path Integral Formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path Integral Formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in the ${\ensuremath{\mu}}_{0}$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\overline{\ensuremath{\mu}}$-scheme of effective dynamics and the continuum limit of the path Integral Formula.
Bogumil Jeziorski - One of the best experts on this subject based on the ideXlab platform.
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exchange splitting of the interaction energy and the multipole expansion of the wave function
Journal of Chemical Physics, 2015Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:The exchange splitting J of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-Integral Formula Jsurf[Φ], the volume-Integral Formula of the symmetry-adapted perturbation theory JSAPT[Φ], and a variational volume-Integral Formula Jvar[Φ]. The calculations are based on the multipole expansion of the wave function Φ, which is divergent for any internuclear distance R. Nevertheless, the resulting approximations to the leading coefficient j0 in the large-R asymptotic series J(R) = 2e(-R-1)R(j0 + j1R(-1) + j2R(-2) + ⋯) converge with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the Jvar[Φ], Jsurf[Φ], and JSAPT[Φ] Formulas are used, respectively. Additionally, we observe that also the higher jk coefficients are predicted correctly when the multipole expansion is used in the Jvar[Φ] and Jsurf[Φ] Formulas. The symmetry adapted perturbation theory Formula JSAPT[Φ] predicts correctly only the first two coefficients, j0 and j1, gives a wrong value of j2, and diverges for higher jn. Since the variational volume-Integral Formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.
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exchange splitting of the interaction energy and the multipole expansion of the wave function
arXiv: Chemical Physics, 2015Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:The exchange splitting $J$ of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-Integral Formula $J_{\textrm{surf}}[\varphi]$, the volume-Integral Formula of the symmetry-adapted perturbation theory $J_{\textrm{SAPT}}[\varphi]$, and a variational volume-Integral Formula $J_{\textrm{var}}[\varphi]$. The calculations are based on the multipole expansion of the wave function $\varphi$, which is divergent for any internuclear distance $R$. Nevertheless, the resulting approximations to the leading coefficient $j_0$ in the large-$R$ asymptotic series $J(R) = 2 e^{-R-1} R ( j_0 + j_1 R^{-1} + j_2 R^{-2} +\cdots ) $ converge, with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the $J_{\textrm{var}}[\varphi]$, $J_{\textrm{surf}}[\varphi]$, and $J_{\textrm{SAPT}}[\varphi]$ Formulas are used, respectively. Additionally, we observe that also the higher $j_k$ coefficients are predicted correctly when the multipole expansion is used in the $J_{\textrm{var}}[\varphi]$ and $J_{\textrm{surf}}[\varphi]$ Formulas. The SAPT Formula $J_{\textrm{SAPT}}[\varphi]$ predicts correctly only the first two coefficients, $j_0$ and $j_1$, gives a wrong value of $j_2$, and diverges for higher $j_n$. Since the variational volume-Integral Formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.
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asymptotics of the exchange splitting energy for a diatomic molecular ion from a volume Integral Formula of symmetry adapted perturbation theory
Physical Review A, 2014Co-Authors: Piotr Gniewek, Bogumil JeziorskiAbstract:The exchange-splitting energy $J$ of the lowest gerade and ungerade states of the ${{\mathrm{H}}_{2}}^{+}$ molecular ion was calculated using a volume Integral expression of symmetry-adapted perturbation theory and standard basis set techniques of quantum chemistry. The performance of the proposed expression was compared to the well-known surface-Integral Formula. Both Formulas involve the primitive function, which we calculated employing either the Hirschfelder-Silbey perturbation theory or the conventional Rayleigh-Schr\"odinger perturbation theory (the polarization expansion). Our calculations show that very accurate values of $J$ can be obtained using the proposed volume-Integral Formula. When the Hirschfelder-Silbey primitive function is used in both Formulas the volume Formula gives much more accurate results than the surface-Integral expression. We also show that using the volume-Integral Formula with the primitive function approximated by Rayleigh-Schr\"odinger perturbation theory, one correctly obtains only the first four terms in the asymptotic expansion of the exchange-splitting energy.
Nengli Lim - One of the best experts on this subject based on the ideXlab platform.
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a stratonovich skorohod Integral Formula for gaussian rough paths
Annals of Probability, 2019Co-Authors: Thomas Cass, Nengli LimAbstract:Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}\mathbf{X}_{t}$, we present a closed-form correction Formula for $\int Y\circ\mathrm{d}\mathbf{X}-\int Y\,\mathrm{d}X$, that is, the difference between the rough and Skorohod Integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-Ito conversion Formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p \frac{1}{3}$. To prove the Formula, we first show that the Riemann-sum approximants of the Skorohod Integral converge in $L^{2}(\Omega)$ by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional Young–Stieltjes Integrals. Next, we append the approximants of the Skorohod Integral with a suitable compensation term without altering the limit, and the Formula is finally obtained after a rebalancing of terms.
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a stratonovich skorohod Integral Formula for gaussian rough paths
arXiv: Probability, 2016Co-Authors: Thomas Cass, Nengli LimAbstract:Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t} = V\left (Y_{t}\right ) \circ \mathrm{d} \mathbf{X}_t$, we present a closed-form correction Formula for $\int Y \circ \mathrm{d} \mathbf{X} - \int Y \, \mathrm{d} X$, i.e. the difference between the rough and Skorohod Integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-It{o} conversion Formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p \frac{1}{3}$. To prove the Formula, we first show that the Riemann-sum approximants of the Skorohod Integral converge in $L^2(\Omega)$ by using a novel characterization of the Cameron-Martin norm in terms of higher-dimensional Young-Stieltjes Integrals. Next, we append the approximants of the Skorohod Integral with a suitable compensation term without altering the limit, and the Formula is finally obtained after a re-balancing of terms.